15-01-IdentityMatrix.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{IdentityMatrix}
\pmcreated{2013-03-22 12:06:29}
\pmmodified{2013-03-22 12:06:29}
\pmowner{mathcam}{2727}
\pmmodifier{mathcam}{2727}
\pmtitle{identity matrix}
\pmrecord{13}{31223}
\pmprivacy{1}
\pmauthor{mathcam}{2727}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-01}
\pmclassification{msc}{15A57}
\pmrelated{KroneckerDelta}
\pmrelated{ZeroMatrix}
\pmrelated{IdentityMap}

\endmetadata

\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
%%%\usepackage{xypic}
\begin{document}
The $n \times n$ \emph{identity matrix} $I$ (or $I_n$) over a ring $R$ (with an identity 1) is the square matrix with coefficients in $R$ given by

$$ I = 
\begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
0 & 0 & \ddots & 0 \\
0 & 0 & \cdots & 1
\end{bmatrix},$$

where the numeral ``1'' and ``0'' respectively represent the multiplicative and additive identities in $R$.  

\subsubsection{Properties}
The identity matrix $I_n$ serves as the multiplicative identity in the ring of $n\times n$ matrices over $R$ with standard matrix multiplication.  For any $n\times n$ matrix $M$, we have $I_nM=MI_n=M$, and the identity matrix is uniquely defined by this property.  In addition, for any $n\times m$ matrix $A$ and $m\times n$ $B$, we have $IA=A$ and $BI=B$.

The $n\times n$ identity matrix  $I$ satisfy the following properties
\begin{itemize}
\item For the determinant, we have $\det I = 1$, and for the trace, we have
$\operatorname{tr}I = n$. 
\item The identity matrix has only one eigenvalue $\lambda =1$ of
multiplicity $n$. The corresponding eigenvectors can be chosen to be 
$v_1=(1,0,\ldots, 0),\ldots, v_n=(0,\ldots, 0,1)$. 
\item The matrix exponential of $I$ gives $e^I = e I$. 
\item The identity matrix is a diagonal matrix. 
\end{itemize}
%%%%%
%%%%%
%%%%%
\end{document}